How to make sense of fractions concretely

Question

I can solve fractions abstractly, for example, $\frac{5}{2}$ divided by $\frac{3}{2}$, you can flip $\frac{3}{2}$ so that $\frac{5}{2}$ multiplied by $\frac{2}{3}$. Specifically, math makes sense abstractly, but concretely it just won't make sense, like in word problems. I understand the concept of complex fractions I know how to solve them, but by applying it on practical use such as a shape it does not make sense. How to make sense of fractions concretely?? or perhaps there is a book that you can advice me that help solve this problem

Answer 1

In general, $\dfrac{a}b$ denotes the quantity $x$, which when multiplied by $b$ gives $a$. In fact, this is the correct way to interpret $\dfrac{a}b$.

In your case, $\dfrac{5/2}{3/2}$ denotes the quantity $x$, which when multiplied by $\dfrac32$ gives us $\dfrac52$, i.e., $$\dfrac32x = \dfrac52 \implies x = \dfrac53$$

Answer 2

Taking your example: $5/2$ divided by $3/2$ is the number of bottles of $3/2$ liters that you need to complete $5/2$ liters. What is the same that how many bottles of $3$ liters are needed to contain $5$ liters. That is, $5/3$ bottles.

Answer 3

So what you would like to know is why $$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a\cdot d}{b\cdot c}$$

to understand this you first need to understand $\frac{1}{x}\cdot x=1$. Knowng this we can see $\frac{c}{d}\cdot\frac{1}{\frac{c}{d}}=1\implies \frac{1}{\frac{c}{d}}=\frac{d}{c}$ (just assume $\frac{c}{d}$ is $x$).

Therefore

$$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot\frac{1}{\frac{c}{d}}=\frac{a}{b}\frac{d}{c}=\frac{ad}{bc}$$

as desired.

Answer 4

This is a playlist of videos especially targeted to understand fractions concretely, hope you enjoy.

Khan academy pre-algebra playlist